Written as a phase-space function, the Hamiltonian (REF ) generates
equations of motion coupling basic expectation values and moments by applying
Hamilton's equations with the Poisson bracket (REF ). However, while
it can easily be seen that \(\lbrace \langle \hat{q}\rangle ,\langle \hat{p}\rangle \rbrace =1\) 
is of canonical form, the moments are not canonical variables. For instance,
\(\lbrace \Delta (q^2),\Delta (p^2)\rbrace =4\Delta (qp)\) . More generally, second-order
moments of \(M\)  classical degrees of freedom have brackets equivalent to the
Lie algebra \({\rm sp}(2M,{\mathbb {R}})\)  [1]}, [2]}, while
higher-order moments have brackets quadratic in moments
[3]}, [4]}.
